Hermann Minkowski (1864-1909)
- Map of Kaliningrad,
from Ann Applebaum's book.
- Major Cities in its hinterland.
He was a professor of mathematics and physics at the University of Zurich, while Einstein was a student there. There he completed the proof that Maxwell's equations are covariant under Lorentz transformations. He published his result in 1908.
Is this the end of the story of the Lorentz covariance of Maxwell's equations? No. This problem was not completely settled until 1990. We shall talk about this next time.
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x2 + y2 + z2 - t2
is a Lorentz-invariant quantity. If we do not change the x and y variables,
-
z2 - t2 = constant.
We are quite familiar with the two-dimensional geometry associated with hyperbolic condition. But this geometry is still is strange to us. I am not the first one to feel in this way. It was Paul A. M. Dirac who invented the light-cone coordinate system to deal with this problem. However, the problem with Dirac was that he never draws pictures in his papers, unlike John A. Wheeler.
Let us write the above formula as
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(t + z) (t - z) =constant.
If we introduce the "light-cone" variables
-
u =(t + z) and v = (t - z),
u v = constant,
I learned this geometry during my high-school years. Since 1973,
most of my papers are based on this squeeze geometry. This geometry
is not only useful for Lorentz-boosted particles, but also provides
the basic mathematical language for
optical sciences.
When my son went to Princeton as a freshman in 1983, Treiman was the
chairman of the department.
He helped my son in many ways. Here is a photo of my family
with the Treimans taken in 1987 at the graduation reception.
As you see, Treiman was a very sharp-looking person. When
I went to Princeton in 1958, many people told me to work for
John A. Wheeler,
but I decided to work for Sam Treiman because he was
so handsome and precise.
Among the many advantages of having Sam Treiman as the advisor, I
was forced to read Steven Weinberg's thesis before writing mine.
Weinberg was not famous at that time. I used this training to
read the papers Weinberg wrote on the Lorentz group in the early
1960s, and to continue my own program on the same subject. I did
enough work to tell the stories Wigner liked to hear in 1985.
The issue was whether Maxwell's equations are consistent with
Wigner's little groups which dictate internal space-time symmetries
of elementary particles. Hermann Minkowski was interested in whether
the Maxwell equations are consistent with Lorentz, not Galilei,
transformations.
Y. S. Kim (2008.10.10)
Sam Treiman was a very important person to me. He was my thesis
advisor at Princeton University. I got my degree in 1961. Four
years before, in 1957, a brilliant man named Steven Weinberg got
his PhD degree under his supervision. This is a photo of Weinberg
and Treiman taken in 1985.
Recent Photos from Lithuania
I went in September (2008) to Vilnius to attend the 12th International
Conferece on Quantum Optics and Quantum Information. As usual, I brought my
camera with me and took a number of photos. I would like share some
of them with you.
copyright@2008 by Y. S. Kim, unless otherwise specified.
The portrait of Hermann Minkowski is from
http://library.thinkquest.org/05aug/01273/whoswho.html.